Abstract

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

1. Introduction

First, we consider the following hyperbolic Navier-Stokes equations [1]:
𝜏𝑢𝑡𝑡+𝑢𝑡−Δ𝑢+∇𝜋+𝑢⋅∇𝑢+𝜏𝑢𝑡⋅∇𝑢+𝜏𝑢⋅∇𝑢𝑡=0,(1.1)div𝑢=0,(1.2)𝑢,𝑢𝑡𝑢(𝑥,0)=0,𝑢1(𝑥),𝑥∈ℝ𝑛,𝑛≥2.(1.3)
Here 𝑢 is the velocity, 𝜋 is the pressure, and 𝜏>0 is a small relaxation parameter. We will take 𝜏=1 for simplicity.

Rack and Saal [1] proved the local well posedness of the problem (1.1)–(1.3). The global regularity is still open. The first aim of this paper is to prove a regularity criterion. We will prove the following theorem.

Theorem 1.1. Let (𝑢0,𝑢1)∈𝐻𝑠+1×𝐻𝑠 with 𝑠>𝑛/2,𝑛≥2 and div𝑢0=div𝑢1=0 in ℝ𝑛. Let (𝑢,𝜋) be a unique strong solution to the problem (1.1)–(1.3). If 𝑢 satisfies
𝑢,∇𝑢,𝑢𝑡∈𝐿1̇𝐵0,𝑇;0∞,∞,(1.5)
then the solution 𝑢 can be extended beyond 𝑇>0.

In our proof, we will use the following logarithmic Sobolev inequality [2]:
‖𝑢‖𝐿∞≤𝐶1+‖𝑢‖̇𝐵0∞,∞log𝑒+‖𝑢‖𝐻𝑠(1.6)
and the following bilinear product and commutator estimates according to Kato and Ponce [3]:
‖Λ𝑠‖(𝑓𝑔)𝐿𝑝≤𝐶‖𝑓‖𝐿𝑝1‖Λ𝑠𝑔‖𝐿𝑞1+‖Λ𝑠𝑓‖𝐿𝑝2‖𝑔‖𝐿𝑞2,(1.7)‖Λ𝑠(𝑓𝑔)−𝑓Λ𝑠𝑔‖𝐿𝑝≤𝐶‖∇𝑓‖𝐿𝑝1‖‖Λ𝑠−1𝑔‖‖𝐿𝑞1+‖Λ𝑠𝑓‖𝐿𝑝2‖𝑔‖𝐿𝑞2,(1.8)
with 𝑠>0, Λ∶=(−Δ)1/2 and 1/𝑝=(1/𝑝1)+(1/𝑞1)=(1/𝑝2)+(1/𝑞2).

Next, we consider the fractional Landau-Lifshitz equation:
𝜕𝑡𝜙=𝜙×Λ2𝛽𝜙𝜙,(1.9)(𝑥,0)=𝜙0(𝑥)∈𝕊2,𝑥∈ℝ𝑛,(1.10)
where 𝜙∈𝕊2 is a three-dimensional vector representing the magnetization and 𝛽 is a positive constant.

When 𝛽=1, using the standard stereographic projection 𝕊2→ℂ∪{∞}, (1.9) can be rewritten as the derivative Schrödinger equation for 𝑤∈ℂ,
𝑖𝑤𝑡+Δ𝑤+4(∇𝑤)21+|𝑤|2𝑤=0.(1.11)

Equation (1.9) is also called the Schrödinger map and has been studied by many authors [4–31]. Guo and Han [32] proved the following regularity criterion:
∇𝜙∈𝐿2(0,𝑇;𝐿∞(ℝ𝑛))(1.12)
with 𝑛≥2.

When 0<𝛽≤1/2, Pu and Guo [33] show the local well posedness of strong solutions and the blow-up criterion
Λ2𝛽𝜙∈𝐿1(0,𝑇;𝐿∞(ℝ𝑛))(1.13)
with 𝑛≤3.

Theorem 1.2. Let 0<𝛽≤1/2. Let 𝑚 be an integer such that 2𝑚>(𝑛+1)/2 for any 𝑛≥1. Let Λ𝛽𝜙0∈𝐻2𝑚 and 𝜙0∈𝕊2 and 𝜙 be a local smooth solution to the problem (1.9) and (1.10). If 𝜙 satisfies
Λ2𝛽𝜙∈𝐿1̇𝐵0,𝑇;0∞,∞(ℝ𝑛)(1.14)
for some finite 𝑇>0, then the solution 𝜙 can be extended beyond 𝑇>0.

Since (𝑢,𝜋) is a local smooth solution, we only need to prove a priori estimates.

First, testing (1.1) by 𝑢 and using (1.2), we see that
𝑑1𝑑𝑡2𝑢2+𝑢𝑢𝑡||||𝑑𝑥+∇𝑢2=𝑢𝑑𝑥2𝑡𝑑𝑥+𝑢⋅∇𝑢⋅𝑢𝑡≤u𝑑𝑥2𝑡1𝑑𝑥+2‖∇𝑢‖𝐿∞‖𝑢‖2𝐿2+‖‖𝑢𝑡‖‖2𝐿2.(2.1)

Testing (1.1) by 4𝑢𝑡 and using (1.2), we find that
𝑑𝑑𝑡2𝑢2𝑡||||+2∇𝑢2𝑢𝑑𝑥+42𝑡𝑑𝑥=−4𝑢⋅∇𝑢+𝑢𝑡𝑢⋅∇𝑢𝑡𝑑𝑥≤𝐶‖∇𝑢‖𝐿∞‖𝑢‖2𝐿2+‖‖𝑢𝑡‖‖2𝐿2.(2.2)

Applying Λ𝑠 to (1.1), testing by Λ𝑠𝑢𝑡 and using (1.2), (1.7), (1.8), and (1.6), we have
12𝑑||Λ𝑑𝑡𝑠+1𝑢||2+||Λ𝑠𝑢𝑡||2||Λ𝑑𝑥+𝑠𝑢𝑡||2𝑑𝑥=−𝑖Λ𝑠𝜕𝑖𝑢𝑖𝑢⋅Λ𝑠𝑢𝑡Λ𝑑𝑥−𝑠𝑢𝑡⋅∇𝑢⋅Λ𝑠𝑢𝑡−𝑑𝑥𝑖Λ𝑠𝜕𝑖𝑢𝑖𝑢𝑡−𝑢𝑖𝜕𝑖Λ𝑠𝑢𝑡Λ𝑠𝑢𝑡𝑑𝑥≤𝐶‖𝑢‖𝐿∞‖‖Λ𝑠+1𝑢‖‖𝐿2‖‖Λ𝑠𝑢𝑡‖‖𝐿2‖‖𝑢+𝐶𝑡‖‖𝐿∞‖‖Λ𝑠+1𝑢‖‖𝐿2+‖∇𝑢‖𝐿∞‖‖Λ𝑠𝑢𝑡‖‖𝐿2‖‖Λ𝑠𝑢𝑡‖‖𝐿2≤𝐶‖𝑢‖𝐿∞+‖∇𝑢‖𝐿∞+‖‖𝑢𝑡‖‖𝐿∞‖‖Λ𝑠+1𝑢‖‖2𝐿2+‖‖Λ𝑠𝑢𝑡‖‖2𝐿2≤𝐶1+‖𝑢‖̇𝐵0∞,∞+‖∇𝑢‖̇𝐵0∞,∞+‖‖𝑢𝑡‖‖̇𝐵0∞,∞log𝑒+‖𝑢‖2𝐻𝑠+1+‖‖𝑢𝑡‖‖2𝐻𝑠⋅‖‖Λ𝑠𝑢𝑡‖‖2𝐿2+‖‖Λ𝑠+1𝑢‖‖2𝐿2.(2.3)

Combining (2.1), (2.2), and (2.3) and using the Gronwall inequality, we conclude that
‖𝑢‖𝐿∞(0,𝑇;𝐻𝑠+1)+‖‖𝑢𝑡‖‖𝐿∞(0,𝑇;𝐻𝑠)≤𝐶.(2.4)

Since 𝜙 is a local smooth solution, we only need to prove a priori estimates. In this section, we denote by (⋅,⋅) the standard 𝐿2 scalar product.

First, testing (1.9) by Λ2𝛽𝜙 and using (𝑎×𝑏)⋅𝑎=0, we see that
12𝑑||Λ𝑑𝑡𝛽𝜙||2𝑑𝑥=0.(3.1)

Testing (1.9) by Δ2𝑚Λ2𝛽𝜙 and using (𝑎×𝑏)⋅𝑎=0, (1.6) and (1.7), we obtain, with (1/𝑝)+(1/𝑞)=(1/𝑝𝛼)+(1/𝑞𝛼)=(1/̃𝑝𝛼)+(1/̃𝑞𝛼)=1/2,
12𝑑||Δ𝑑𝑡𝑚Λ𝛽𝜙||2=𝑑𝑥𝜙×Λ2𝛽𝜙,Δ2𝑚Λ2𝛽𝜙=Δ𝑚𝜙×Λ2𝛽𝜙,Δ𝑚Λ2𝛽𝜙=Δ𝑚𝜙×Λ2𝛽𝜙+2𝑚−1𝛼=1𝐶𝛼𝐷2𝑚−𝛼𝜙×Λ2𝛽𝐷𝛼𝜙,Δ𝑚Λ2𝛽𝜙=Λ𝛽Δ𝑚𝜙×Λ2𝛽𝜙+2𝑚−1𝛼=1𝐶𝛼𝐷2𝑚−𝛼𝜙×Λ2𝛽𝐷𝛼𝜙,Δ𝑚Λ𝛽𝜙‖‖Λ≤𝐶2𝛽𝜙‖‖𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖2𝐿2+𝐶‖Δ𝑚𝜙‖𝐿𝑝‖‖Λ3𝛽𝜙‖‖𝐿𝑞‖‖Δ𝑚Λ𝛽𝜙‖‖𝐿2+2𝑚−2𝛼=1𝐶𝛼‖‖𝐷2𝑚−𝛼𝜙‖‖𝐿𝑝𝛼‖‖Λ3𝛽𝐷𝛼𝜙‖‖𝐿𝑞𝛼+‖‖𝐷2𝑚−𝛼Λ𝛽𝜙‖‖𝐿̃𝑝𝛼‖‖Λ2𝛽𝐷𝛼𝜙‖‖𝐿̃𝑞𝛼‖‖Δ𝑚Λ𝛽𝜙‖‖𝐿2‖‖Λ≤𝐶2𝛽𝜙‖‖𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖2𝐿2‖‖Λ≤𝐶1+2𝛽𝜙‖‖̇𝐵0∞,∞‖‖Δlog𝑒+𝑚Λ𝛽𝜙‖‖𝐿2‖‖Δ𝑚Λ𝛽𝜙‖‖2𝐿2,(3.2)
which yields
‖‖Λ𝛽‖‖𝜙(𝑡)𝐻2𝑚≤𝐶.(3.3)
Here we have used the following interesting Gagliardo-Nirenberg inequalities:
‖Δ𝑚𝜙‖𝐿𝑝‖‖Λ≤𝐶2𝛽𝜙‖‖1−𝜃0𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖𝜃0𝐿2with𝑝=2𝑚−𝛽𝑚−𝛽,𝜃0=2𝑚−2𝛽,‖‖Λ2𝑚−𝛽3𝛽𝜙‖‖𝐿𝑞‖‖Λ≤𝐶2𝛽𝜙‖‖𝜃0𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖1−𝜃0𝐿2with𝑞=2(2𝑚−𝛽)𝛽,‖‖𝐷2𝑚−𝛼𝜙‖‖𝐿𝑝𝛼‖‖Λ≤𝐶2𝛽𝜙‖‖1−𝜃𝛼𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖𝜃𝛼𝐿2with𝜃𝛼=2𝑚−𝛼−2𝛽2𝑚−𝛽,𝑝𝛼=4𝑚−2𝛽,‖‖Λ2𝑚−𝛼−2𝛽3𝛽𝐷𝛼𝜙‖‖𝐿𝑞𝛼‖‖Λ≤𝐶2𝛽𝜙‖‖𝜃𝛼𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖1−𝜃𝛼𝐿2with𝑞𝛼=4𝑚−2𝛽,‖‖𝐷𝛼+𝛽2𝑚−𝛼Λ𝛽𝜙‖‖𝐿̃𝑝𝛼‖‖Λ≤𝐶2𝛽𝜙‖‖1−̃𝜃𝛼𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖̃𝜃𝛼𝐿2̃𝜃with𝛼=2𝑚−𝛼−𝛽2𝑚−𝛽,̃𝑝𝛼=4𝑚−2𝛽,‖‖Λ2𝑚−𝛼−𝛽2𝛽𝐷𝛼𝜙‖‖𝐿̃𝑞𝛼‖‖Λ≤𝐶2𝛽𝜙‖‖̃𝜃𝛼𝐿∞‖‖Δ𝑚Λ𝛽𝜙‖‖1−̃𝜃𝛼𝐿2with̃𝑞𝛼=4𝑚−2𝛽𝛼.(3.4)
This completes the proof.